Extending of Edge Even Graceful Labeling of Graphs to Strong r -Edge Even Graceful Labeling

Edge even graceful labeling of a graph G with p vertices and q edges is a bijective f from the set of edge E G to the set of positive integers 2,4 , … , 2 q such that all the vertex labels f ∗ V G , given by f ∗ u = ∑ u v ∈ E G f u v mod 2 k , where k = max p , q , are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to r -edge even graceful labeling and strong r -edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an r -edge even graceful graph. Furthermore, the minimum number r for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an r -edge even graceful labeling was found. Finally, we proved that the even cycle C 2 n has a strong 2 -edge even graceful labeling when n is even.